Bake lights (lamps) to vertex color
No, I don't think you can directly, as far as I know, but what you could probably do instead is a multi-step process to achieve your desired final result.

You can only bake to a texture, so bake said lamp influence to an image based texture, then use said texture to create a new vertex color layer from it using an addon to translate from image to vertex colors.

Rough steps: Create your desired mesh, add a "neutral" material to it, like say a purely diffuse white material that will receive clean colors from lamps.

Unwrap said mesh to a new UV Map, create a new texture and bake that object to said texture.

Save the image, then use the addon UV/Bake Texture to Vertex Colors to create a new Vertex Color layer based on said texture.